The idea is to consider the higher geometric quantization not just of the low codimension transgressions, but of all transgressions of cconn\mathbf{c}_{conn}. The basic constructions that higher geometric quantization is concerned with are indicated in the following table. All of them have also a fundamental interpretation in twistedcohomology (independent of any interpretation in the context of quantization) this is indicated in the right column of the table:

But for any further nontrivial such autoequivalence in the slice we would need in particular a gauge transformation parameterized by (2k+1)(2k+1)-forms over test manifolds from C∧dCC \wedge d C to itself. But the only closed 2k2k-forms that we can produce naturally from CC are multiples of C∧CC \wedge C. But these all vanish since CC is of odd degree 2k+12k+1.

then diagonal gauge transformations B(G×G)conn→B(G×G)conn\mathbf{B}(G \times G)_{conn}
\to \mathbf{B}(G \times G)_{conn} have interesting extensions to quantomorphisms, because for g:U→Gg : U \to G the given gauge transformation at stage of definition UU, the Chern-Simons form transforms by an exact term